1.
Schwinger parametrisation:
\(\langle \! \langle \mathcal {O}_1({p}_1) \ldots \mathcal {O}_n({p}_n) \rangle \! \rangle = \Big (\prod _{i<j}^n\int _0^\infty v̥_{ij}\, v_{ij}^{-\alpha _{ij}-1}\Big ) f(\hat{{v}})\,\mathcal {U}^{-d/2} e^{-\mathcal {F}/\mathcal {U}}\) 
Here, the \(v_{ij}^{-d/2}\)  factors in (REF ) cancel with those associated with \(\mathcal {U}^{-d/2}\)  via (REF ).
2.
Lee-Pomeransky parametrisation [1]}:
\(\langle \! \langle \mathcal {O}_1({p}_1) \ldots \mathcal {O}_n({p}_n) \rangle \! \rangle =\Big (\prod _{i<j}^n\int _0^\infty v̥_{ij}\, v_{ij}^{-\alpha _{ij}-1} \Big )f(\hat{{v}})\,(\mathcal {U}+\mathcal {F})^{-d/2}\) 
3.
Feynman parametrisation:
\(\langle \! \langle \mathcal {O}_1({p}_1) \ldots \mathcal {O}_n({p}_n) \rangle \! \rangle =\Big (\prod _{i<j}^n\int _0^\infty v̥_{ij}\,v_{ij}^{-\alpha _{ij}-1} \Big )\delta \Big (1-\sum _{i<j}^n \kappa _{ij}v_{ij}\Big ) f(\hat{{v}})\,\mathcal {U}^{\omega -d/2} \mathcal {F}^{-\omega }\) 
where \(\omega =(n-1)d/2+\sum _{i<j}^n\alpha _{ij} = (-\Delta _t+(n-1)d)/2\)  and the constants \(\kappa _{ij}\ge 0\)  can be chosen arbitrarily provided they are not all zero.The Feynman parametrisation follows from the Schwinger parametrisation by setting \(v_{ij}=y_{ij}/\sigma \)  subject to the constraint \(\sum _{i<j}^n \kappa _{ij}y_{ij}=1\) . The \(\mathcal {U}\)  and \(\mathcal {F}\)  are homogeneous polynomials of weights \((n-1)\)  and \((n-2)\)  respectively, meaning that \(\mathcal {F}(v_{ij})/\mathcal {U}(v_{ij}) =\sigma \mathcal {F}(y_{ij})/\mathcal {U}(y_{ij})\)  while the Jacobian can be evaluated as per appendix B of [2]}. We then perform the scale integral over \(\sigma \)  and relabel the \(y_{ij}\rightarrow v_{ij}\) .
If we choose all \(\kappa _{ij}=1\)  then the integration region is a simplex in the space spanned by the \(v_{ij}\) . Alternatively, we can set a single \(\kappa _{ij}\)  to unity and the rest to zero which trivialises one of the integrations at the cost of obscuring permutation invariance.

